I was recently given a promotion by an online gambling website for £10 free credit which I could withdraw after having wagered at least £200 in total. What's the best betting strategy for maximising my EV when playing roulette? The non trivial feature of this problem being that, if I lose my initial £10 I cannot continue gambling to achieve the long term expected value of my bets.
I decided at first I was continually betting £0.50 on red. Reasonably quickly I decided to go with a different strategy of betting £0.50 on red and £0.50 black each time meaning I'd break even as long as the ball did not land on 0.
How does one calculate the EV in each scenario, simply betting on a single a colour, or betting on both black and red. It's trivially simple if you have unlimited money, and in both cases the expected value is 36/37~0.973 (since we're playing european roulette with just one green square) however in my case, if I go through a bad streak and lose my £10 I can not continue. Therefore it seems the EV becomes partially dependent on the variance of the strategy.
Secondly what produces a higher EV for both the strategies above, a large amount of small bets, or a smaller amount of larger bets. I.e. one thousand £0.50 bets or one hundred £5?
My gut instinct says that because of this feature that once I'm down a total of £10 I can't continue makes me think that smaller bets are produce a higher expected value but I don't know how to prove it.
Edit: EV stands for expected value, however I may be using the term incorrectly. To explain what I mean here's a simple example of two strategies which are easy to calculate. One where you bet £5 on red and £5 on black every bet, your mean returns is £10*(36/37)^20=~£5.8 i.e. you need to avoid hitting a 0, 20 times.
If instead you bet all the money you have on red every time, you need to win 5 times in a row before you've wagered at least £200, so your mean returns is £320*(18/37)^5=~£8.7 which is greater than £5.8. This is what I meant by expected value. So not the expected value of an individual bet, but the expected value of your winnings after betting at least £200.
Or you could think of it as, if a million people all got the same promotion and then pooled their winnings and shared them out between them, whats the best strategy they should all follow.